无穷维哈密顿系统中的极大环面：一种重整化群方法
Maximal tori in infinite-dimensional Hamiltonian systems: a Renormalization Group approach
AI总结：
针对无穷多弱相互作用旋转子系统的不变环面存续问题，采用基于树形式的重整化群方法，证明只需少量技术调整即可将有限维的相关方案推广到无穷维情形，为可积系统微扰的解析性要求提供了简单案例的讨论。
摘要：
我们研究了由无穷多个彼此弱相互作用的旋转子构成的力学系统中无穷维不变环面的存在性。我们明确考虑仅依赖于角度的相互作用，旨在在一个简单情形下讨论可积系统的微扰所需满足的解析性条件，以确保一大类具有有限能量的正测度不变环面的存续性。我们给出的不变环面存续性证明采用了基于树形式体系的重整化群（Renormalization Group）方案——即以树的形式对运动方程的解进行图形化表示——该方案已被广泛应用于有限维问题中。该方法十分有效且灵活：在确定了函数空间设定后，它只需少量技术性调整就能自然推广到无穷维情形。
英文摘要：
We study the existence of infinite-dimensional invariant tori in a mechanical system of infinitely many rotators weakly interacting with each other. We consider explicitly interactions depending only on the angles, with the aim of discussing in a simple case the analyticity properties to be required on the perturbation of the integrable system in order to ensure the persistence of a large measure set of invariant tori with finite energy. The proof we provide of the persistence of the invariant tori implements the Renormalization Group scheme based on the tree formalism -- i.e. the graphical representation of the solutions of the equations of motion in terms of trees -- which has been widely used in finite-dimensional problems. The method is very effectual and flexible: it naturally extends, once the functional setting has been fixed, to the infinite-dimensional case with only minor technical-natured adaptations.
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